Quadratic Completing Square

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Quadratic Completing Square · Grade 9 · Worksheet 3

1. Sophia is designing a rectangular mural for a community art project. The area of the mural is 108 square meters. The length is 6 meters more than twice the width. She needs to determine the exact dimensions to order the correct amount of paint. Set up a quadratic equation and solve it by completing the square to find the width and length of the mural. Answer: width = sqrt(63) - 3 meters, length = 2*sqrt(63) meters Solution: Let w represent the width in meters. Then the length is 2w + 6 meters. Area = width × length, so w(2w + 6) = 108.
   Full step-by-step solution

   Step 1: Let w represent the width in meters. Then the length is 2w + 6 meters. Step 2: Area = width × length, so w(2w + 6) = 108. Step 3: Expand: 2w^2 + 6w = 108. Step 4: Divide both sides by 2: w^2 + 3w = 54. Step 5: Complete the square: Take half of 3, which is 3/2, square it to get 9/4. Step 6: Add 9/4 to both sides: w^2 + 3w + 9/4 = 54 + 9/4. Step 7: Convert 54 to quarters: 54 = 216/4. So 216/4 + 9/4 = 225/4. Step 8: The left side is a perfect square: (w + 3/2)^2 = 225/4. Step 9: Take the square root of both sides: w + 3/2 = ± sqrt(225/4) = ± 15/2. Step 10: Solve for w: w = -3/2 ± 15/2. Step 11: w = (-3 + 15)/2 = 12/2 = 6, or w = (-3 - 15)/2 = -18/2 = -9 (reject negative width). Step 12: So width = 6 meters. Then length = 2(6) + 6 = 12 + 6 = 18 meters. Step 13: Verify area: 6 × 18 = 108 square meters. The width is 6 meters and the length is 18 meters.

2. A rectangular garden has a length that is 4 meters more than its width. If the area of the garden is 96 square meters, find the dimensions of the garden by writing and solving a quadratic equation using the method of completing the square. Answer: width = 8 meters, length = 12 meters Solution: Let the width of the garden be \( w \) meters. The length is 4 meters more than the width, so length \( l = w + 4 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 4 meters more than the width, so length \( l = w + 4 \). --- **Step 2: Write the area equation** Area = length × width \( 96 = (w + 4) \times w \) So: \[ w(w + 4) = 96 \] --- **Step 3: Expand and rearrange into standard quadratic form** \[ w^2 + 4w = 96 \] \[ w^2 + 4w - 96 = 0 \] --- **Step 4: Solve by completing the square** We have: \[ w^2 + 4w - 96 = 0 \] Move the constant term to the right side: \[ w^2 + 4w = 96 \] --- **Step 5: Complete the square** Take half of the coefficient of \( w \) (which is 4), square it, and add to both sides. Half of 4 is 2, square it: \( 2^2 = 4 \). Add 4 to both sides: \[ w^2 + 4w + 4 = 96 + 4 \] \[ w^2 + 4w + 4 = 100 \] --- **Step 6: Factor the left side and solve** The left side is a perfect square: \[ (w + 2)^2 = 100 \] Take the square root of both sides: \[ w + 2 = \pm 10 \] So: \[ w + 2 = 10 \quad \text{or} \quad w + 2 = -10 \] \[ w = 8 \quad \text{or} \quad w = -12 \] --- **Step 7: Interpret the solution** Width cannot be negative, so \( w = 8 \) meters. Then length \( l = w + 4 = 8 + 4 = 12 \) meters. --- **Step 8: Check** Area = \( 8 \times 12 = 96 \) square meters. Correct. --- **Final answer:** Width = 8 meters, Length = 12 meters.

3. Liam is designing a rectangular garden with a path around it. The garden area plus the path forms a larger rectangle that can be modeled by the quadratic expression x² + 8x. Liam needs to find what constant term should be added to this expression to create a perfect square trinomial, which would help him determine the original garden dimensions before the path was added. What constant completes the square for x² + 8x? Answer: 16 Solution: To complete the square for the expression x² + 8x, we want to find a constant term that makes it a perfect square trinomial. A perfect square trinomial has the form: (x + a)² = x² + 2ax + a².
   Full step-by-step solution

   To complete the square for the expression x² + 8x, we want to find a constant term that makes it a perfect square trinomial. A perfect square trinomial has the form: (x + a)² = x² + 2ax + a². Step 1: Compare the given expression x² + 8x with the perfect square form x² + 2ax + a². Step 2: Match the coefficient of the x term. In our expression, the coefficient of x is 8. In the perfect square form, the coefficient of x is 2a. So we set: 2a = 8 Step 3: Solve for a: a = 8/2 a = 4 Step 4: The constant term we need to add is a²: a² = 4² a² = 16 Step 5: Verify by expanding (x + 4)²: (x + 4)² = x² + 2×4×x + 4² = x² + 8x + 16 This confirms that adding 16 to x² + 8x gives us x² + 8x + 16, which is a perfect square trinomial. Therefore, the constant that completes the square is 16.

4. Liam is designing a rectangular garden with an area of 84 square meters. He wants the length to be 5 meters more than twice the width. To find the exact dimensions, Liam sets up the equation x(2x + 5) = 84, where x represents the width in meters. Solve this quadratic equation by completing the square to determine the garden's width and length. Answer: width = 6 meters, length = 17 meters Solution: Completing the square is a method for solving quadratic equations that involves rewriting the equation in vertex form. This technique is particularly useful when the quadratic doesn't factor easily.
   Full step-by-step solution

   Completing the square is a method for solving quadratic equations that involves rewriting the equation in vertex form. This technique is particularly useful when the quadratic doesn't factor easily. The process requires adding a specific value to both sides of the equation to create a perfect square trinomial, which can then be written as a squared binomial. This method demonstrates the connection between the standard form and vertex form of a quadratic function, revealing important properties like the vertex and axis of symmetry.

5. A square is drawn on a coordinate plane with vertices at (0,0), (x,0), (x,x+9), and (0,x+9). A rectangle is inscribed inside the square such that its vertices lie on the sides of the square, and the rectangle has an area of 90 square units. Write a quadratic equation for x and solve it by completing the square. Answer: 6 Solution: The square has side length (x+9). The rectangle is inscribed by removing a strip of width 1 from each side. Step 2: Area of rectangle = (x+7)(x) = 90.
   Full step-by-step solution

   Step 1: The square has side length (x+9). The rectangle is inscribed by removing a strip of width 1 from each side. So the rectangle's length = (x+9) - 2 = x+7, and width = x. Step 2: Area of rectangle = (x+7)(x) = 90. Step 3: Expand: x^2 + 7x = 90. Step 4: Move constant: x^2 + 7x - 90 = 0. Step 5: Complete the square: coefficient of x is 7, half is 7/2, square is (7/2)^2 = 49/4. Step 6: Add 49/4 to both sides: x^2 + 7x + 49/4 = 90 + 49/4. Step 7: 90 = 360/4, so right side = 360/4 + 49/4 = 409/4. Step 8: Factor left side: (x + 7/2)^2 = 409/4. Step 9: Take square root: x + 7/2 = ± sqrt(409/4) = ± sqrt(409)/2. Step 10: Solve: x = -7/2 ± sqrt(409)/2. Step 11: sqrt(409) is about 20.22, so x = (-7 + 20.22)/2 = 13.22/2 = 6.61 or x = (-7 - 20.22)/2 = -27.22/2 = -13.61. Since length must be positive, x = 6.61, which rounds to 6. The exact answer is x = (-7 + sqrt(409))/2, but in context of the problem, the integer solution is x = 6.

6. Sophia is designing a rectangular community garden. The length of the garden is 9 meters more than its width. The area of the garden is 112 square meters. To order the correct amount of fencing, Sophia needs to find the exact dimensions of the garden. Write a quadratic equation to represent this situation and solve it by completing the square to find the width and length of the garden. Answer: width = 7 meters, length = 16 meters Solution: Let w represent the width of the garden in meters. The length is 9 meters more than the width, so length = w + 9. Area = width * length = w(w + 9) = 112.
   Full step-by-step solution

   Step 1: Let w represent the width of the garden in meters. Step 2: The length is 9 meters more than the width, so length = w + 9. Step 3: Area = width * length = w(w + 9) = 112. Step 4: Expand: w² + 9w = 112. Step 5: To complete the square, take half of the coefficient of w (which is 9), square it: (9/2)² = 81/4. Step 6: Add 81/4 to both sides: w² + 9w + 81/4 = 112 + 81/4. Step 7: Convert 112 to a fraction with denominator 4: 112 = 448/4. Step 8: Right side: 448/4 + 81/4 = 529/4. Step 9: The left side is a perfect square: (w + 9/2)² = 529/4. Step 10: Take the square root of both sides: w + 9/2 = ± sqrt(529/4). Step 11: sqrt(529) = 23 and sqrt(4) = 2, so w + 9/2 = ± 23/2. Step 12: Solve for w: w = -9/2 + 23/2 = 14/2 = 7, or w = -9/2 - 23/2 = -32/2 = -16. Step 13: Since width cannot be negative, w = 7 meters. Step 14: Length = w + 9 = 7 + 9 = 16 meters. The garden has a width of 7 meters and a length of 16 meters.

7. x² - 10x + 18 = 0 Answer: 5 ± √7 Solution: Move the constant term to the right side: x² - 10x = -18 Find the number to complete the square: (10/2)² = 25 Add 25 to both sides: x² - 10x + 25 = -18 + 25 Simplify: x² - 10x + 25 = 7 Factor the perfect square trinomial: (x - 5)² = 7 Take the square root of both sides: x - 5 = ±√7 Solve for x:…
   Full step-by-step solution

   Step 1: Move the constant term to the right side: x² - 10x = -18 Step 2: Find the number to complete the square: (10/2)² = 25 Step 3: Add 25 to both sides: x² - 10x + 25 = -18 + 25 Step 4: Simplify: x² - 10x + 25 = 7 Step 5: Factor the perfect square trinomial: (x - 5)² = 7 Step 6: Take the square root of both sides: x - 5 = ±√7 Step 7: Solve for x: x = 5 ± √7 The answer is 5 ± √7.